I would argue that Hedda experiences both a modern and a classical tragedy. A classical tragic hero is a man or woman of high status who experiences a downfall because of a tragic flaw in his/her personality. Audiences are meant to learn from the experience of the tragic hero, to feel pity for the fallen and learn moral lessons.
Modern tragedy tends to implement social flaws as part of the tragic trajectory that the characters experience. Arthur Miller used this idea in both his modern tragedies, The Crucible and Death of a Salesman. In the first, John Proctor is flawed, but it is the society that persecutes him. In the second, Willy Loman makes many mistakes, but he is led astray by visions of the American Dream as enforced by his society.
In a classic way, Hedda is a woman of high status with a tragic flaw. Much reference is made of her father, the general: "General Gabler's daughter! Think of the sort of life she was accustomed to in her father's time" (1.12-3). She actually marries a bit beneath her, more for the security of being married than for love. This is part of her tragic flaw. She acts more out of impulse than rational thought and she is selfish. She likes her husband because he idolizes her. However, he bores her, and being married to him increases her angst and her need to act out.
In a modern sense, her tragedy is a result of social convention. She is not a nice person, granted. However, she is a woman of the Victorian Era, and this provides some needed explanation. Women in Hedda's time had no agency. They were secondary citizens, restricted in their rights and judged by their every action. If found to be lacking in any way, a woman was shunned from society, which most often put her in a desperate financial difficulties. The last line is "But, good God! People don't do such things." In closing, Ibsen focuses on social propriety. The strict laws and behaviors enforced in this era lead to the death of Hedda in a very real way, making her death tragic.